we factor terms of a generalized Fibonacci sequence a[n] = a[n-23] + a[n-24] , initialized with only one 1, in Pari/GP.
allocatemem(16*10^6)
addprimes(...)
addprimes(...)
len=24; a=List([]); for(i=2,len,listput(a,0)); listput(a,1); for(i=1,8410,x=a[len]+a[len-1]; listpop(a); listinsert(a,x,1); gettime; y=factorint(x); t=gettime; print("out ",i," time ",t," ",x," ",y))
the two addprimes commands are given below. they are the 2nd and 3rd largest prime factors, respectively, of the most difficult terms to factor among the first 8410 terms. with both addprimes, the computation takes about 3 hours, down from the original several months of computing.
without allocatemem, pollard rho phase runs out of memory on term 6524 (GP/Pari Version 2.9.4).
here is a function that prints out any given term by index. (it could be made more efficient, exponentiating a matrix.)
jumpto(n) = my(a,i,x,len); a=List([]); len=24; for(i=1,len-1,listput(a,0)); listput(a,1); for(i=1, +oo, x=a[len]+a[len-1]; listpop(a); listinsert(a,x,1); if(i==n,break)); x
for reference, here are the first 531 terms of this sequence:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 10 10 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 15 20 15 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 21 35 35 21 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 8 28 56 70 56 28 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 9 36 84 126 126 84 36 9 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 10 45 120 210 252 210 120 45 10 1 0 0 0 0 0 0 0 0 0 0 0 0 1 11 55 165 330 462 462 330 165 55 11 1 0 0 0 0 0 0 0 0 0 0 0 1 12 66 220 495 792 924 792 495 220 66 12 1 0 0 0 0 0 0 0 0 0 0 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 0 0 0 0 0 0 0 0 0 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 0 0 0 0 0 0 0 0 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 0 0 0 0 0 0 0 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 0 0 0 0 0 0 1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1 0 0 0 0 0 1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1 0 0 0 0 1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1 0 0 0 1 20 190 1140 4845 15504 38760 77520 125970 167960 184756 167960 125970 77520 38760 15504 4845 1140 190 20 1 0 0 1 21 210 1330 5985 20349 54264 116280 203490 293930 352716 352716 293930 203490 116280 54264 20349 5985 1330 210 21 1 0 1 22 231 1540 7315 26334 74613 170544 319770 497420 646646 705432 646646 497420 319770 170544 74613 26334 7315 1540 231 22 1 1 23
the asymptotic growth rate of this lag-24 sequence is 1.029939697 (FindRoot[0==1+x-x^24, {x,1}, WorkingPrecision->10]). log(2)/log(x) = 23.496 ; log(10)/log(x) = 78.1 .
for our arbitrarily chosen cutoff for factorizations taking too long (20-30 seconds, depending on computer), the first term for which induced an entry to addprimes was term 5185, 216 bits, 65 digits, which factors into 5 * (32-digit prime) * (33-digit prime). the 32-digit prime is 35883310089669970781089136839889, the first entry in the addprimes commands below.
no two terms induced duplicated entries to our addprimes lists.
the final term done, the 8410th, was 354 bits, 107 digits, and was the most time-consuming factorization: 257 hours. 11 * (49-digit prime) * (58-digit prime) . the 49-digit prime is 1506291006111119117205211837954859047440030121999 , the final entry in the first addprimes command below (second largest prime factors).
goal was factorization of a random-ish collection of numbers of many different sizes. in retrospect, there were better ways to sample a geometrically growing random set of integers (future post tykznwfp). previously, factoring random numbers 128-256 bits.
the small numbers scattered among the addprimes lists below, some as small as 18 digits, represent instances of the elliptic curve method being unlucky. future work: identify where ECM was lucky.
\\ second largest prime factors, 884 primes
addprimes([35883310089669970781089136839889, 153505251588962425237, 65638786155986809, 6335151847969010091037, 83760219098371044014977, 108580968225874351874999416991, 25148004015165244945393, 346874642039053903, 141905839647428613187, 155950072352408546336434929029, 31695074109394258271, 7075949561986955161111, 125483384255560494440256649, 409560837590847550933746593, 12473249520758578507143390773231, 2810264938578751397147, 17385895598475846559133959, 19006523816425424693, 16206094373800117798177637, 2501912890817549778663555882541, 71551955299832764348843, 191065982914609136798378183, 33042768649358639137672369, 1240706693728224290647, 81155427312671365848286210846009, 43163174962449998794753514311, 110035050983875374718606602113, 26307854174268046560883297205454731, 2420787406638495969223, 439423190482007024750843389, 162357542812059374124689948099, 4203003726922737668816771, 34877148612585284194278833, 191338612293976252796780621209, 66842928216881960549977905644131, 5161861367102510064212163797, 417892158488830677096314973979, 159300167253469173179, 107898937199391510017500103, 3016347922027349373419373167, 123666967923718613633, 17655251116100366309, 1203491122519025797, 4613211177207116570971, 19723720405744196003, 194878241936273643855517523, 40653974502953972903137, 94026581851621796717, 40988261561385405099932107, 18871835948325629576963893, 552622890722408992857229519, 1146734147003352508341824410891, 464218964329931177186826001, 19325463480002724973102829, 185131052392684573047972238889219, 704219834788168094084743, 360105654228634026854133871, 35144238726511396515401, 58353547941244794347, 238243554125224366348028382559, 39876464302417338577036103647, 1017476210558008476833, 203402490652059275985826939, 866989487436993264259049535061, 6608967975692051760082730839, 66973401804976882714049357, 650105045402896843952273, 1974649814742640599115901633328797, 5338046368915062952759217, 13531402087340150879216419315519, 743764443065750615064397, 2280015094453148244653, 4081421335632058138897492132181, 2489197589701671349, 6702390421178620686281, 1140370359494428247938622546349683, 8855293109537402154509, 38446847658423228196343, 364889503687309647501600549649547, 873741357802646848032187961911, 9752949346243296231811, 41308466177712936023944784254027, 85464556443785093, 361335558416328913013438957, 160690400832726349514142713, 23432601626315867, 41529121437393908664342499, 22261274218525066162222862907218057, 205391996226575615407, 1527655959295362914173082609, 661558506163603653397801, 5081184794696793315681503, 1944170353778077410960584901653, 60043778859191504003362410173, 2628170515316281810423480175599, 223678765464483589658779, 3639576972170554849508095601, 19902449794680151551443786796634757, 1540753250181253612070147716339, 239684517499807765054038961, 3374878201013512482766667, 3525855290745536644687859203, 284235995965269093828800099, 149579326541296101390489450329111, 15641810332203294019951, 173667712823578344209242231928131, 1924630846143174370239659452957, 2383004447765599568322454254397, 311944908026286851, 844023876970433615543, 2588266488728386930818224878283109853, 1549375183978647196943742959627, 95961836785874982136067453456471, 50922135911096107408117049, 47791666744379988463510230754057, 42710349504521258305820052211, 27114912709565088403948453, 3188221413877264553939412099839, 131326053847804569152758575330791, 385757482876481726501, 90102216412371430926861941, 429146086666430796946308293, 530547325948505522193805171, 153856507570538281234181, 168857542989049482459103, 14683486239752177614829, 5161827257408654722511977057, 332462535739503406258419260832239, 33624436710443037576569095781899, 914795309714143746365951, 997494115250597025961961, 58344607021256458975011394207, 803998489011048627969735607, 172636793427747629005570277, 8846506736153187752655899, 442835399372669800254280711057607, 55631414814783489646969480388248496701, 1442444510961613852817990077, 3212165308500447730666515431489147, 5503137884444636306224433, 54166242077332034608129026095203, 4842893494044004359133915303, 7431165711739942869601741, 6420460859384758037260815890683, 136671282868298358848891025353079243983, 871462660121042952962743582651, 91354844304282074600448336292103033713, 347771791214109926109463, 54802097354616633434697539514209, 140022872608738886604592886641, 108937285167552944843455824352581739, 2384061519099881653738156849, 6879330955267671109103, 100598712949879180066157, 22265304911646173453571259174008372419813, 607814192622859935017515917229, 165581786573241387429781, 1488325233857100929288154811, 186423992743441023996049, 307730963774079817894031, 1735758232010784023888411, 166379280512578165294916153, 523139845158004481492863, 131830440065868720078283, 2973433744285732447910913703821551, 67061346279547306138445909, 136996063484078854337155563828218284229, 1037176865141656123713303004631, 885967681963076310749, 47922441186914985233, 22930859609123598922980077, 390969596197593615342653, 240320452086552278431597, 15958267839658131535801399372777, 286316911705026911992574788261930859, 37477253347495984286299600451, 11966109851192479625908149551, 4809808791257642410489125373, 1887627648082311436098139, 6435957759614472319, 211511854005771607555009, 85977932470638309842653967, 80843867967554827734928759, 89796451811415988626490175581, 3897813258342946773825988012246668143, 11269415900608682254116893601301367, 30110300314646907503, 10034799545048863011286323693233, 4344318680657622521345221771, 5777991916510513627296759322921007, 4182001258276275642649039845959, 921561239704253584210429499893, 756661704960482446966063, 52235499951374127730627321, 128616786702360134938431153620200761623, 3373939617681324936107, 1896548581445534867480827841023042095761, 868643344872479348237211337969, 543734900819272346561884889, 22722627437188979271541, 157960573236092703995677067273231, 13892358884749929685668522889217754738157, 13909817747505572273, 47131452342179671180534373, 117088324288704235631009117, 562029965967075806303, 6323863428884693022162966773, 48343215099300749666783310287959, 6595574760921953501220751593457, 1446895022568931426117335679447, 106904929311577916095331, 173352915648297197175007, 6403471970134703903, 240652019453800653474494161, 470900224298406252365549487024553, 17550741045672163676579, 166644645131221497203, 1539937318644616316507, 83027490999511593555211, 79444446773167411889, 97055550232347657379916647, 11480392486727455456717, 166371821130542178114452665073, 132920420876154559635518160341848931909, 10463703112223400390475463989487, 1067823602987332237247057, 723825574673307664484750917, 109141706745011649809, 1939213815838953321329079256446547, 50463982824348025417253, 12678119149100743400451349, 3625720310314251085103, 6327081263150458177705754921507, 14623520896916829675866983, 2730113561836606187697853, 522923328881073510167741347722851983, 56236846768392642046665013, 135854441250155471919063121607, 37797194180329014416429, 2304814157787285735588973329941, 4842209641580282719415699293, 4101692701586040006189713, 770365612728908836626312017, 11283709456532905026184339, 1153504277887012780393387296153361, 12187214811100302067110139, 43621816193965972890791994447017, 232681262031802957515298251288079, 255208943676890762518766462156947, 17600700964464556692385844263, 5301091324548809371285999609, 173974779718007120346017543, 6577623264048732178605011, 40649840033814571635173171, 2847126448036660093177521898399, 522345345881346732683, 209696457419579308571, 913878722749048980866709311250833, 1300278765965918280768991, 1744672074219865452295801351, 131775522164313275071325503, 104176952207104466475940790248177, 6408680102224474015017863111827, 21564917223530556259828417057619, 437496265771852169, 592859578708579260516902754193, 10581844894410039059967931, 1702247874065147040451, 63009656240536348144171, 6608449951425047711810639850220690669, 5645862913448749839432496303897507, 3467028549826624619757938362288091, 8234068779856298543634371185701254587, 34158311100191370467947487, 471958895439992338782642023, 59371065121384561678489797181701313, 2782824786840696841379707, 222622602099593970306998786222921, 12038290883270552281, 9176005280455208981671194863354197, 100513363315083059017951759, 149401849743030716317, 438456215459144864836253797, 117034103830863217833151294129039, 1475963780941091432839672594177151, 26836683751018181551691406341923, 60288709415185194278677, 439429147756871154648757, 32236938873448200208769398347227, 6743823091967885673561983, 7694699168068156129, 22530815998857279987782304144837857, 8801759897034751833272341, 420588224681033838971, 14568637705657498887358640653741, 5454545251824726757159, 374871220985324771269, 8863639101682509009538494878395607, 426555152359925631771529, 685533198952864122281, 132336208695067236264178257023, 5747762695605677631187644690521344229, 17862970951494907430852187698494198663, 5428755736101910605035301041, 8063440676826363842254971792697883, 17711848635873043111974647633, 4334182393626145798721, 637770386366142517223905438680301073, 63044715504247762767761, 9321767037697845683983688237687, 297579775708088499863411, 95759296264696105949, 180878909381375037271900533220116138307, 10037583285565567853760367, 1029933589893039071354707526136823307, 3258819054529941139264123403, 5031474200869084431805030287618671, 158664753903407378173, 456771276204267675428742825445369, 1422246560811626160584774407, 3282518178996641820461367157, 9389357041019051757176652862035697813, 9373147824580014750547, 333007201330915136966752223716595971, 999812166309283060745461909, 930673071840027943976585901546151, 420161481101669645723417015875169367251, 834354759244296226739115151031, 1048815696915202751228323845167, 32359639890055384751357360964162181, 2292553716056596592512558577, 24732087155106229312765831, 87098630216478436988797, 2209681081550049846574261, 2862716212203663780941, 121160653368080357412924760853, 76145329689486974244730714396459, 138513157184580209750285443, 50637181918065549146481992472091, 942467262593951323682716399046708522317, 580875894185119153827248699, 5070172343870660873401, 2124591069063730197943, 14199417523908927881444178421, 8883260271140956142679473371, 650417804596071044263081597312153, 12307108246639242300867776293, 201832153502026487427656509, 1083163751023366157699923573, 973747486295626729978616572523, 317487378200027218633, 225090236867882298559, 142952508522288220589683, 95758119248509685870839, 14354029427276297314847, 330156450099310307299, 12143401308482790179, 5958558216922879485931323539803115083, 355716003500919416423466902107, 1290718568930333315298660085043441, 686530451677268829357880891, 408936469005017921029, 175445724709201112857392146514424148239, 317360002351425484530121960189, 2950413490938589016486861, 339746507922044224546960498583, 64167532843269930242571551, 199850586871860672889, 5479370602154804442595391447, 99577804230583938402278717, 3783417794327744831182119132503774401481, 13716906991084595101262445575417, 210423165484804063078512118483, 287070536075729935712849893, 33354392518318350723403854648143603509, 4455416911295415107296760509, 6220744783686863373582165802351, 77078322001219834207453, 6319593289955487901664395373, 33041929410150928981527384029964716674657, 7491173285235961568422360237769, 40445781138099519448309, 5947825634217541676603317, 28821651286846446851399, 24361789764729067653204987377, 813905048046717450176200484449, 1266288693403558091384040799, 46196879599825181676896554747897, 9675909456748904779033291, 3262947943349521498056870257141589437, 464894071046782109854568374267523, 787694135491846132487064899, 273348465798208158621011050639, 33761132480132651221, 71894209226574036464690514313149437, 104700357430550162423, 246267016315135628104811081411739309941, 7932016655177758283753, 30926614350445225564440867947, 213412037011161451578860857743503, 63712303701792202235626061, 2447571327002012697723534023, 834822597761616506778361, 2241468911765114607311511229271, 3241975471471717141243193123, 20055648608999842837709, 4292576066236367802776131, 268216279863408780232922633, 2179845071386053446916529807723, 18989063737458776761015808985667, 10121521691862486959, 1094676847874303054029, 302777780803178373541, 28800750111265828110469, 47970993386990426958633227, 710434704998695207684709, 12630594675776662675597295556427883, 202108642204054895996889509, 126515872780531077203, 200938574198573328778685263, 11868250454875698311401, 748671626217620383954421033014079, 123780594388332199960026850828177, 663998937632753439815992754213, 17688711174421497924612277, 114452888743843605591791159, 58560103926017224782367588479218509733, 14104039112867371127557, 4367498550454211613743581, 327463917139191581351, 249854599271727784716539, 9248157882323843825522814922284429843871, 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\\ third largest prime factors, 44 primes
addprimes([59576718681627821, 98501123617245949, 138598798273932761, 5408979750324745901, 6462339564524872397, 20652875228488335017, 9163858315338862231, 356120421609113092153889, 6932943755493033953, 3354908076703056902939, 127948735601176844742109, 44161085023388023319, 225215623122648550052863, 20398034995569724763, 20611044148501285500533, 97586618754794392073, 26002217302461336127, 7785570944466431887, 8247006860941500257, 1655733203483381773445239, 2953069114940846260159, 374514648625536459379, 15059370546976585199536492065499, 643928788451607137, 33014245889922881, 8257029439741656395243, 7203813202136662187929, 1571908746467275931927, 94531644301368913421, 699184400849008886446013, 13525372779622820625647803208993, 93700146536472538850512807, 55253357281568457877, 464529691551452966431, 61962374346345162428939, 2973647431499584536625367, 1091823859608267013505531, 58349173047963275430283, 51694948793834014911443, 2021225824713875422650053219, 195067149129775878515527871831, 349793956030892222094153407, 4126129372002305119, 24511366861314949223]);
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